Parallel Lines – Same Gradient, Never Meeting
Two lines are parallel if they have exactly the same gradient. Geometrically, this means they point in exactly the same direction and, as long as they're not the same line, they will never cross – no matter how far you extend them in either direction.
Given raw coordinate points rather than ready-made equations, checking whether two lines are parallel becomes a two-step calculation: work out each line's gradient using the gradient formula, then simply compare the two numbers.
Checking for Parallel Lines
Lines are parallel when their gradients are equal. Given points, compute each gradient with m = (y₂ − y₁) ÷ (x₂ − x₁), then compare.
Gradient of A = (5 − 1) ÷ (2 − 0) = 2. Gradient of B = (4 − 0) ÷ (3 − 1) = 2. Both gradients are 2, so the lines are parallel.
Parallel lines share the gradient, so m = 4. Substitute (2, 15): 15 = 4(2) + c, so c = 15 − 8 = 7. The line is y = 4x + 7.
Real-Life Application
- Railway tracks: the two rails run perfectly parallel to stay a constant distance apart.
- Road markings: lane lines are drawn parallel to keep lanes a consistent width.
- Architecture: parallel structural beams distribute weight evenly.
Key Takeaways
- Parallel lines have equal gradients and never intersect.
- Compute gradients from points using the gradient formula, then compare.
- A parallel line through a new point shares the same m but has a different c.
Practice: Parallel Lines
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